It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.
The faces of this honeycomb's cells form four families of parallel planes, each with a 3.6.3.6 tiling.
John Horton Conway calls this honeycomb a truncated tetrahedrille, and its dual oblate cubille.
The quarter cubic honeycomb can be constructed in slab layers of truncated tetrahedra and tetrahedral cells, seen as two trihexagonal tilings.
The reflection generated form represented by its Coxeter-Dynkin diagram has two colors of truncated cuboctahedra.